IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v180y2024ics0960077924000973.html
   My bibliography  Save this article

Multiple bifurcations of a time-delayed coupled FitzHugh–Rinzel neuron system with chemical and electrical couplings

Author

Listed:
  • Hu, Dongpo
  • Ma, Linyi
  • Song, Zigen
  • Zheng, Zhaowen
  • Cheng, Lifang
  • Liu, Ming

Abstract

In this paper, a time-delayed coupled FitzHugh–Rinzel neuron system (two neurons) with electrical and chemical couplings is discussed. First, when choosing different strengths of electrical (or chemical) coupling, the number and position of equilibria of the neuron system without time delays affected by the variation of chemical (or electrical) coupling are investigated in detail. Next, based on the discussion about the equilibria of the coupled neuron system without time delays, the codimension one bifurcations of equilibria or limit cycles including static bifurcation, Hopf bifurcation and fold bifurcation of limit cycles are deduced when choosing the strength of electrical coupling or chemical coupling as the bifurcation parameter, respectively. Furthermore, the codimension two bifurcations of equilibria are discussed which exhibit more fascinating and complicated dynamical behaviors. Synchronization problems influenced by the strength of electrical coupling are also considered which contain the complete synchronous and asynchronous state. After the discussion of the neuron system without time delays, the impact of time delays on the stability of symmetric equilibria of coupled FitzHugh–Rinzel neuron system is explored. The phenomenon of stability switching by a Hopf bifurcation is well detected. Delay-dependent Hopf curves are found and the dynamical behaviors near the intersection point of Hopf bifurcation (Hopf–Hopf bifurcation point) in the parameter plane of two time delays are investigated where there exists the coexistence of two limit cycles with different frequencies. The numerical simulations are exhibited after the discussion of each part.

Suggested Citation

  • Hu, Dongpo & Ma, Linyi & Song, Zigen & Zheng, Zhaowen & Cheng, Lifang & Liu, Ming, 2024. "Multiple bifurcations of a time-delayed coupled FitzHugh–Rinzel neuron system with chemical and electrical couplings," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    • Handle: RePEc:eee:chsofr:v:180:y:2024:i:c:s0960077924000973
    DOI: 10.1016/j.chaos.2024.114546
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077924000973
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2024.114546?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Duan, Lixia & Lu, Qishao, 2006. "Codimension-two bifurcation analysis on firing activities in Chay neuron model," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1172-1179.
    2. Semenov, Vladimir V. & Bukh, Andrei V. & Semenova, Nadezhda, 2023. "Delay-induced self-oscillation excitation in the Fitzhugh–Nagumo model: Regular and chaotic dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    3. Robert Rosenbaum & Jonathan Rubin & Brent Doiron, 2012. "Short Term Synaptic Depression Imposes a Frequency Dependent Filter on Synaptic Information Transfer," PLOS Computational Biology, Public Library of Science, vol. 8(6), pages 1-18, June.
    4. Li, Bo & Liang, Houjun & He, Qizhi, 2021. "Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    5. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Analysis and control of the bifurcation of Hodgkin–Huxley model," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 247-256.
    6. Bo Yan & Fatemeh Parastesh & Shaobo He & Karthikeyan Rajagopal & Sajad Jafari & Matjaå½ Perc, 2022. "Interlayer And Intralayer Synchronization In Multiplex Fractional-Order Neuronal Networks," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(10), pages 1-11, December.
    7. Yu, Dong & Lu, Lulu & Wang, Guowei & Yang, Lijian & Jia, Ya, 2021. "Synchronization mode transition induced by bounded noise in multiple time-delays coupled FitzHugh–Nagumo model," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    8. Simone Holler & German Köstinger & Kevan A. C. Martin & Gregor F. P. Schuhknecht & Ken J. Stratford, 2021. "Structure and function of a neocortical synapse," Nature, Nature, vol. 591(7848), pages 111-116, March.
    9. Xu, Quan & Tan, Xiao & Zhu, Dong & Bao, Han & Hu, Yihua & Bao, Bocheng, 2020. "Bifurcations to bursting and spiking in the Chay neuron and their validation in a digital circuit," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yu, Xihong & Bao, Han & Chen, Mo & Bao, Bocheng, 2023. "Energy balance via memristor synapse in Morris-Lecar two-neuron network with FPGA implementation," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    2. Xiao, Fangli & Fu, Ziying & Jia, Ya & Yang, Lijian, 2023. "Resonance effects in neuronal-astrocyte model with ion channel blockage," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Shadizadeh, S. Mohadeseh & Nazarimehr, Fahimeh & Jafari, Sajad & Rajagopal, Karthikeyan, 2022. "Investigating different synaptic connections of the Chay neuron model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    4. Lin, Yi & Liu, Wenbo & Hang, Cheng, 2023. "Revelation and experimental verification of quasi-periodic bursting, periodic bursting, periodic oscillation in third-order non-autonomous memristive FitzHugh-Nagumo neuron circuit," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    5. Chen, Xiongjian & Wang, Ning & Wang, Yiteng & Wu, Huagan & Xu, Quan, 2023. "Memristor initial-offset boosting and its bifurcation mechanism in a memristive FitzHugh-Nagumo neuron model with hidden dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    6. Yu, Dong & Wang, Guowei & Ding, Qianming & Li, Tianyu & Jia, Ya, 2022. "Effects of bounded noise and time delay on signal transmission in excitable neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    7. Xu, Quan & Wang, Yiteng & Chen, Bei & Li, Ze & Wang, Ning, 2023. "Firing pattern in a memristive Hodgkin–Huxley circuit: Numerical simulation and analog circuit validation," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    8. Fateev, I. & Polezhaev, A., 2024. "Chimera states in a lattice of superdiffusively coupled neurons," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    9. Yu, Dong & Wu, Yong & Yang, Lijian & Zhao, Yunjie & Jia, Ya, 2023. "Effect of topology on delay-induced multiple resonances in locally driven systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    10. Wu, Huagan & Gu, Jinxiang & Guo, Yixuan & Chen, Mo & Xu, Quan, 2024. "Biphasic action potentials in an individual cellular neural network cell," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    11. Lu, Qishao & Yang, Zhuoqin & Duan, Lixia & Gu, Huaguang & Ren, Wei, 2009. "Dynamics and transitions of firing patterns in deterministic and stochastic neuronal systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 577-597.
    12. Chen, Yixuan & Guo, Qun & Zhang, Xiaofeng & Wang, Chunni, 2024. "Numerical approach and physical description for a two-capacitive neuron and its adaptive network dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 189(P2).
    13. Li, Fangyuan & Chen, Zhuguan & Bao, Han & Bai, Lianfa & Bao, Bocheng, 2024. "Chaos and bursting patterns in two-neuron Hopfield neural network and analog implementation," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    14. José Moya-Díaz & Ben James & Federico Esposti & Jamie Johnston & Leon Lagnado, 2022. "Diurnal changes in the efficiency of information transmission at a sensory synapse," Nature Communications, Nature, vol. 13(1), pages 1-15, December.
    15. Hu, Xueyan & Ding, Qianming & Wu, Yong & Huang, Weifang & Yang, Lijian & Jia, Ya, 2024. "Dynamical rewiring promotes synchronization in memristive FitzHugh-Nagumo neuronal networks," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    16. Virginia Gao & Julita Chlebowicz & Karlton Gaskin & Juan A. Briano & Lauren E. Komer & André Pineda & Shrey Jhalani & Saad Ahmad & Eseosa Uwaifo & Luca S. Black & Jillian E. Haller & Serge Przedborski, 2025. "Synaptic vesicle-omics in mice captures signatures of aging and synucleinopathy," Nature Communications, Nature, vol. 16(1), pages 1-25, December.
    17. Ding, Shoukui & Wang, Ning & Bao, Han & Chen, Bei & Wu, Huagan & Xu, Quan, 2023. "Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: Dynamics analysis and circuit implementation," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    18. Zhou, Shu & Cheng, Zebang & Huang, Guodong & Zhu, Rui & Chai, Yuan, 2024. "Synchronization evaluation of memristive photosensitive neurons in multi-neuronal systems," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    19. Li, Bo & Liang, Houjun & Shi, Lian & He, Qizhi, 2022. "Complex dynamics of Kopel model with nonsymmetric response between oligopolists," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    20. Lei Zhang, 2025. "Study of Educational Information Resource Download Quality with Optimal Symmetrical Interval Solution of Fuzzy Relation Inequality in the Format of a System of Differential Equations," Mathematics, MDPI, vol. 13(10), pages 1-22, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:180:y:2024:i:c:s0960077924000973. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.